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The BEKK GARCH class of models presents a popular set of tools for applied analysis of dynamic conditional covariances. Within this class the analyst faces a range of model choices that trade off flexibility with parameter parsimony. In the most flexible unrestricted BEKK the parameter dimensionality increases quickly with the number of variables. Covariance targeting decreases model dimensionality but induces a set of nonlinear constraints on the underlying parameter space that are difficult to implement. Recently, the rotated BEKK (RBEKK) has been proposed whereby a targeted BEKK model is applied after the spectral decomposition of the conditional covariance matrix. An easily estimable RBEKK implies a full albeit constrained BEKK for the unrotated returns. However, the degree of the implied restrictiveness is currently unknown. In this paper, we suggest a Bayesian approach to estimation of the BEKK model with targeting based on Constrained Hamiltonian Monte Carlo (CHMC). We take advantage of suitable parallelization of the problem within CHMC utilizing the newly available computing power of multi-core CPUs and Graphical Processing Units (GPUs) that enables us to deal effectively with the inherent nonlinear constraints posed by covariance targeting in relatively high dimensions. Using parallel CHMC we perform a model comparison in terms of predictive ability of the targeted BEKK with the RBEKK in the context of an application concerning a multivariate dynamic volatility analysis of a Dow Jones Industrial returns portfolio. Although the RBEKK does improve over a diagonal BEKK restriction, it is clearly dominated by the full targeted BEKK model.

Existing multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models either impose strong restrictions on the parameters or do not guarantee a well-defined (positive-definite) covariance matrix. I discuss the main multivariate GARCH models and focus on the BEKK model for which it is shown that the covariance and correlation is not adequately specified under certain conditions. This implies that any analysis of the persistence and the asymmetry of the correlation is potentially inaccurate. I therefore propose a new Flexible Dynamic Correlation (FDC) model that parameterizes the conditional correlation directly and eliminates various shortcomings. Most importantly, the number of exogenous variables in the correlation equation can be flexibly augmented without risking an indefinite covariance matrix. Empirical results of daily and monthly returns of four international stock market indices reveal that correlations exhibit different degrees of persistence and different asymmetric reactions to shocks than variances. In addition, I find that correlations do not always increase with jointly negative shocks implying a justification for international portfolio diversification.

This chapter examines the dynamic linkages between the returns of Bitcoin, gold, and oil by using daily closing price data between July 17, 2010 and January 8, 2021. This study applies the diagonal BEKK–GARCH model for the purpose of analyzing a volatility spillover of variables in positive or negative ways. The empirical results show that the lagged returns inversely affect their current returns in oil. Based on the return spillovers between Bitcoin and gold, the empirical results indicate a unidirectional return spillover from Bitcoin to gold. Moreover, the authors found a unidirectional return transmission is observed from oil to Bitcoin, implying that oil returns are useful in forecasting Bitcoin returns. These findings are not only valuable for understanding of the interrelationships between the returns of Bitcoin, gold, and oil, but they are also of great interest to portfolio managers, investors, and investment funds that are actively dealing in Bitcoin, gold, and oil.

This study aims to analyze the volatility spillover effects of crude oil, gold price, interest rate (yield) and the exchange rate (USD (United States Dollar)/INR (Indian National Rupee)) on inflation volatility in India.

This study uses the multivariate Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models (Baba, Engle, Kraft and Kroner [BEKK]-GARCH and dynamic conditional correlation [DCC]-GARCH) to examine the volatility spillover effect of macroeconomic indicators and strategic commodities on inflation in India. The monthly data are collected from January 2000 till December 2020 for the crude oil price, gold price, interest rate (5-year Indian bond yield), exchange rate (USD/INR) and inflation (wholesale price index [WPI] and consumer price index [CPI]).

In BEKK-GARCH, the results reveal that crude oil price volatility has a long time spillover effect on inflation (WPI). Furthermore, no significant short-term volatility effect exists from crude oil market to inflation (WPI). However, the short-term volatility effect exists from crude oil to inflation while considering CPI as inflation. Gold price volatility has a bidirectional and negative spillover effect on inflation in the case of WPI. However, there is no price volatility spillover effect from gold to inflation in the case of CPI. The price volatility in the exchange rate also has a negative spillover effect on inflation (but only on CPI). Furthermore, volatility of interest rates has no spillover effect on inflation in WPI or CPI. In DCC-GARCH, a short-term volatility impact from all four macroeconomic indicators to inflation is found. Only crude oil and exchange rate have long-term volatility effect on inflation (CPI).

In an economy, inflation management is an essential task. The findings of the current study can be beneficial in this endeavor. The knowledge of the volatility spillover effect of all the four markets undertaken in the study can be significantly helpful in inflation management, especially for inflation-targeting policy.

It is observed that no other study has addressed this issue. We do not find any other research which studies the volatility spillover effect of gold, crude oil, interest rate and exchange rate on the inflation volatility. The current study is novel with a significant contribution to the vast knowledge in this context.

This paper aims to employ GARCH-class models (GARCH, IGARCH and CGARCH) to estimate the volatility persistence on crude oil, US, Gulf Corporation Council (GCC), Brazil, Russia, India and China (BRIC) stock markets. Also, the paper investigates the volatility spillover and the dynamic conditional correlation between crude oil, US stock index and stock indices of GCC and BRIC countries. The results prove a high degree of volatility persistence in the crude oil and stock markets. Based on the BEKK-GARCH and DCC-GARCH results, the paper finds strong evidence of the contagion effect of the oil shock and US financial crisis of 2008 on GCC and BRIC stock markets.

In the beginning, the paper uses univariate GARCH models to estimate the volatility persistence of the oil market, US stock market, and GCC and BRIC stock markets. Then, the paper uses a trivariate BEKK-GARCH model of Malik and Hammoudeh to examine the volatility spillover between oil market, US stock market and stock markets for GCC and BRIC countries. Finally, the paper analyses the dynamic conditional correlation between US market and each stock market of GCC and BRIC countries using the DCC-GARCH model. Also, the paper estimates the dynamic conditional correlation between oil market and all stock markets.

The results indicate the contagion effect of the oil shock and US financial crisis of 2008 on the GCC stock markets which are among the most important oil-exporting countries and also on BRIC stock markets which are among the emergent countries which are characterized by high economic growth level.

The contribution of this paper is to investigate the existence of contagion effect between oil market, US stock market and two panels of emerging stock markets which have different economic characteristics, the GCC and BRIC countries, during the oil shock and US financial crisis period of 2008-2009.

The nexus of commodity prices with inflation is one of the main concerns for a nation's economy like India. The literature does not have enough volatility-based study, especially using the multivariate GRACH family of models to find a link between these two. It is the main reason for the conduct of this study. This paper aims to estimate the volatility effects of commodity prices on inflation.

For ten years (2011–2022), future prices of selected seven agriculture commodities and inflation indices (wholesale price index [WPI] and consumer price index [CPI]) are gathered every month. BEKK GARCH model (BGM) and DCC GARCH model (DGM) are employed to determine the volatility effect of commodity prices (CPs) on inflation.

The authors find that volatility's short-term (shock) impact on agricultural CPs to inflation does not exist. However, the long-term volatility spillover effect (VSE) is significant from commodities to inflation.

The study's findings have a significant implication for the policymakers to take a long-term view on inflation management regarding commodity prices. The findings can facilitate policy on the choice of commodities and the flexibility of their trading on the commodities derivatives market.

The findings of the study are unique. The authors do not observe any study on the volatility effect of agri-commodities (agricultural commodities) prices on inflation in India. This paper applies advanced techniques to provide novel and reliable evidence. Hence, this research is believed to contribute significantly to the knowledge body through its novel evidence and advanced approach.

This paper assesses the co-movement between Pax gold and six fiat currencies. It also investigates the optimal time-varying hedge ratios in order to examine the properties of Pax gold as a diversifier and hedge asset.

This paper examines the volatility spillover between Pax gold and fiat currencies using the framework of wavelet analysis, BEKK-GARCH models and Range DCC-GARCH. Moreover, this paper proposes to use the covariance and variance structure obtained from the new range DCC-GARCH framework to estimate the time-varying optimal hedge ratios, the optimal weighs and the hedging effectiveness.

Wavelet coherence method reveals that, at low frequency, large zone of co-movements appears for the pairs Pax gold/EUR, Pax gold/JPY and Pax gold/RUB. Further, the BEKK results show unidirectional (bidirectional) transmission effects between Pax gold and EUR, GBP, JPY and CNY (INR, RUB) fiat currencies. Moreover, the Range DCC results show that the Pax gold and the fiat currency returns are weakly correlated with low coefficients close to zero. Thus, Pax gold seems to serve as a safe haven asset against the systematic risk of fiat currency markets. In addition, the results of optimal weights show that rational investor should invest more in Pax gold and less in fiat currencies. Concerning the hedge ratios results, the findings reveal that the INR (JPY) fiat currency appears to be the most expensive (cheapest) hedge for the Pax-gold market. However, the JPY’s fiat currency appears to be the cheapest one. As for hedging effectiveness results, the authors found that hedging strategies including fiat currencies–Pax gold pairs are most likely to sharply decrease the portfolio’s risk.

A comprehensive understanding of the relationship between Pax Gold and fiat currencies is crucial for refining portfolio strategies involving cryptocurrencies. This research underscores the significance of grasping volatility transmissions between these currencies, providing valuable insights to guide investors in their decision-making processes. Moreover, it encourages further exploration into the interdependencies of digital currencies. Additionally, this study sheds light on effective contagion risk management, particularly during crises such as Covid-19 and the Russia–Ukraine conflict. It underscores the role of Pax Gold as a safe-haven asset and offers practical guidance for adjusting portfolios across various economic conditions. Ultimately, this research advances our comprehension of Pax Gold’s risk-return profile, positioning it as a potential hedge during periods of uncertainty, thereby contributing to the evolving literature on cryptocurrencies.

This study’s primary value lies in its pioneering empirical examination of the time-varying correlations and scale dependence between Pax Gold and fiat currencies. It goes beyond by determining optimal time-varying hedge ratios through the innovative Range-DCC-GARCH model, originally introduced by Molnár (2016) and distinguished by its incorporation of both low and high prices. Significantly, this analysis unfolds within the unique context of the Covid-19 pandemic and the Russian–Ukrainian conflict, marking a novel contribution to the field.

The purpose of this paper is to analyze the nature of returns and volatility spillovers between exchange rates and stock price in the IBSA nations (India, Brazil, South Africa).

The study uses VAR framework and the recently proposed Spillover measure of Diebold and Yilmaz to examine the returns and volatility spillover between exchange rates and stock prices of IBSA nations. In addition, multivariate GARCH with time varying variance‐covariance BEKK model is used as a benchmark against the spillover methodology proposed by Diebold and Yilmaz.

The results of multivariate GARCH model suggests the integration between stock and foreign exchange markets and indicates the existence of bi‐directional volatility spillover between stock and foreign exchange markets in the IBSA countries. Spillover results using the Diebold Yilmaz model suggest the bi‐directional contribution between stock and foreign exchange market, in terms of both returns and volatility spillovers. Overall, results confirm the presence of returns and volatility spillovers within the IBSA nations and, in particular, the stock markets play a relatively more important role than foreign exchange markets in the first and second moment interactions and spillovers.

The market participants may consider the relationship between the exchange rate and stock index to predict the future movement of each other effectively. Multinational companies interested in exchange rate forecasting may consider the stock market as an important attribute. There is an interesting implication for portfolio managers too because of the spillover stock and foreign exchange markets. This knowledge would help to create a fund which performs well. Moreover, the paper can help regulators and policy makers in IBSA nations to understand the structure of the market in a better way and then design their policies.

The study contributes to the literature by extending the existing studies on the spillover between stock price and exchange rate by investigating the issue for three emerging economies, India, Brazil and South Africa. Unlike most studies in the literature which focus on multivariate GARCH model, this is the first study which explores the issue of returns and volatility spillover between the stock prices and the exchange rates using spillover measure of Diebold and Yilmaz and much longer and recent daily data. Moreover, multivariate GARCH with time varying variance‐covariance BEKK model is used as a benchmark against the spillover methodology proposed by Diebold and Yilmaz.

The purpose of this paper is to analyze the existence of volatility spillover effect in frontier markets. This study also examines whether any linkages exist among these markets or not.

Monthly data of regional frontier markets, from 2009 to 2016, are analyzed using Multivariate GARCH (BEKK and Dynamic Conditional Correlation (DCC)) models.

The result of cointegration test shows that the sample frontier markets are not linked in long run, and Granger causality test reveals that the markets under consideration do not cause each other even in the short run. BEKK test says that the effect of the arrival of shock from the own market does not last for longer, whereas shock from other markets lasts with the stronger persistence, and according to DCC test, the volatility spillover exists for all the markets.

The results of present study suggest that the frontier markets are not cointegrated in the long run as well as in the short run, which opens the doors for long-term investments in these markets in future, which may lead to decent returns. Long-term investors may draw the benefits from including the financial assets in their portfolios from these non-integrated frontier markets; nevertheless, they have to consider and implement diversification and hedging strategies during the period of financial turmoil, so as to protect themselves against economic and financial distress.

Significant work has been done on developed, developing and emerging markets but frontier markets are not explored much so far. This paper is an attempt to see the status of frontier stock markets as potential financial markets for diversification benefits.